This pull-down of the conjugation connecting the neighboring Hilbert
spaces is an elementary consequence of definition (REF )
which
played just a marginal role in Ref. [1]} at \(K=3\) 
(cf. equation Nr. 10 in
loc. cit.). The importance of relation (REF )
grows with the growth of \(K\) .
At any \(K\ge 3\) 
it leads to the
closed-form definition
\(\Lambda ^{\ddagger (j)} =\Theta _{(K-1,j)}^{-1}\,\Lambda ^{\dagger }\,\Theta _{(K-1,j)}\,,\ \ \ \ \ \Theta _{(K-1,j)}=Z_{{K-1}}\,Z_{{K-2}}\,\ldots \,Z_{j+2}\,Z_{j+1}\,\) 
